The invention relates to a positioning device for machining a workpiece fastened to a rotatable support, which device is provided with a longitudinal carriage movable in an x-direction, on which is present a transverse carriage which is movable relative to the longitudinal carriage in a y-direction perpendicular to the x-direction, while a .phi.-carriage is present on the transverse carriage and capable of rotating relative to the transverse carriage about an axis of rotation which is perpendicular to the x-direction and y-direction.
A positioning device of the type described above and functioning as a universal lathe is known from U.S. Pat. No. 2,051,127. In this known positioning device, a holder for a machining tool is movable relative to the .phi.-carriage along a straight guide which is present on the .phi.-carriage. The .phi.-carriage is attached to the transverse carriage and is rotatable relative to the latter about a vertical axis of rotation by means of a swiveling-plate present on the transverse carriage. The transverse carriage is movable relative to the longitudinal carriage in the y-direction along a straight guide rail attached to the longitudinal carriage. A drawback of the known lathe is that an x-position and y-position of a tool tip, once set, will change if the .phi.-carriage is rotated about the axis of rotation. If a desired x-position and y-position of the tool tip are to be maintained upon rotation of the .phi.-carriage, the two positions have to be corrected by means of a calculated displacement of the longitudinal carriage and the transverse carriage, which generally does not benefit the positioning accuracy of the tool tip. A further drawback of the known lathe is the stacking of the various carriages, which adversely affects the rigidity of the construction, and thus the positioning accuracy of the tool. In addition, an inaccuracy in the movement of the longitudinal carriage or the transverse carriage in a direction parallel to the axis of rotation always leads to a deviation in the tool position in the said direction.